I wrote a small recursive function to convert a list (a0 a1 ... an) coefficients into a polynomial function
;;Given a list (a0 a1 ... an) return a function that computes
;;p(x) = a0 + a1*x + ... + an*x^n
(define (polynomial coeffs)
(lambda (x)
(cond
[(= (length coeffs) 0) 0]
[(= (length coeffs ) 1) (first coeffs)]
[else (+ (first coeffs) (* x ((polynomial (rest coeffs)) x)))])))
and it seems to work fairly well. The problem is that it only works for numeric functions. If I define an operator (i.e., a function, usually linear,
that maps functions to functions) such as the difference operator
(delta f)(x) = f(x) - f(x - 1)
or
(define (delta f)
(lambda (x)
(- (f x) (f (- x 1)))))
and p is polynomial, I ought to be able to compute a new operator
p(delta) = a0 + a1*delta + a2*delta^2 + ... + an*delta^n
(where delta^k is just delta composed with itself or, if you prefer, applied k times).
Now, my question is: is there a notation in Racket for representing composition that I should use, or am I better off writing a function that maps an operator
A to p(A) where p is a polynomial?